Topological embeddings into random 2-complexes

Michael Farber, Tahl Nowik

Research output: Contribution to journalArticlepeer-review

Abstract

We consider 2-dimensional random simplicial complexes Y in the multiparameter model. We establish the multiparameter threshold for the property that every 2-dimensional simplicial complex S admits a topological embedding into Y asymptotically almost surely. Namely, if in the procedure of the multiparameter model on n vertices, each i-dimensional simplex is taken with probability pi = pi(n), then the threshold is (Formula presented.). Our claim in one direction is in fact slightly stronger, namely, we show that if (Formula presented.) is sufficiently larger than (Formula presented.) then every S has a fixed subdivision S′ which admits a simplicial embedding into Y asymptotically almost surely. In the other direction we show that if (Formula presented.) is sufficiently smaller than (Formula presented.), then asymptotically almost surely, the torus does not admit a topological embedding into Y.

Original languageEnglish
Pages (from-to)664-675
Number of pages12
JournalRandom Structures and Algorithms
Volume58
Issue number4
DOIs
StatePublished - Jul 2021
Externally publishedYes

Keywords

  • multiparameter threshold
  • random simplicial complexes

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